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Chattering and its approximation in control of psoriasis treatment

  • * Corresponding author: Ellina Grigorieva

    * Corresponding author: Ellina Grigorieva 
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  • We consider a nonlinear system of differential equations describing a process of the psoriasis treatment. Its phase variables are the concentrations of T-lymphocytes, keratinocytes and dendritic cells. A scalar bounded control is introduced into this system to reflect the medication dosage. For such a control system, on a given time interval the minimization problem of the Bolza type functional is stated. Its terminal term is the concentration of keratinocytes at the terminal time, and its integral term is the product of the non-negative weighting coefficient with the total cost of the psoriasis treatment. This cost is linear in the control and proportional to the concentration of keratinocytes. For the analysis of such a problem, the Pontryagin maximum principle is used. As a result of this analysis, it is shown that if the weighting coefficient is zero, then the corresponding optimal control can contain a singular arc. We establish that it is a chattering control, and therefore does not make much sense as a type of a medical treatment. If the weighting coefficient is positive, then the corresponding optimal control is bang-bang, and it can be presented as a type of psoriasis treatment. In addition, when this coefficient tends to zero, such optimal controls can be considered as chattering approximations. Therefore, the convergence of these optimal controls, the corresponding optimal solutions of the original system, and the minimum values of the functional are studied. The obtained theoretical results are illustrated by numerical calculations and the corresponding conclusions are made.

    Mathematics Subject Classification: Primary: 49N90, 90C90; Secondary: 93C95.


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  • Figure 1.  Optimal solutions and optimal control for $ \xi = 0 $ (Case B): upper row: $ l_{0}^{*}(t) $, $ k_{0}^{*}(t) $; lower row: $ m_{0}^{*}(t) $, $ u_{0}^{*}(t) $

    Figure 2.  Optimal solutions and optimal control for $ \xi = 0 $ (Case B): upper row: $ l_{0}^{*}(t) $, $ k_{0}^{*}(t) $; lower row: $ m_{0}^{*}(t) $, $ u_{0}^{*}(t) $

    Figure 3.  Optimal solutions and optimal control for $ \xi = 0.001 $: upper row: $ l_{\xi}^{*}(t) $, $ k_{\xi}^{*}(t) $; lower row: $ m_{\xi}^{*}(t) $, $ u_{\xi}^{*}(t) $

    Figure 4.  Optimal solutions and optimal control for $ \xi = 0.000001 $: upper row: $ l_{\xi}^{*}(t) $, $ k_{\xi}^{*}(t) $; lower row: $ m_{\xi}^{*}(t) $, $ u_{\xi}^{*}(t) $

    Table 1.  Minimum values $ J_{\xi}^{*} $ of the functional $ J_{\xi}(u) $ and the corresponding numbers of iterations $ M_{\xi} $

    $ \xi = 10.0 $ $ \xi = 1.0 $ $ \xi = 0.1 $ $ \xi = 0.01 $
    $ J_{\xi}^{*} = 3.030960 $ $ J_{\xi}^{*} = 3.030960 $ $ J_{\xi}^{*} = 3.016830 $ $ J_{\xi}^{*} = 2.905979 $
    $ M_{\xi} = 69 $ $ M_{\xi} = 72 $ $ M_{\xi} = 80 $ $ M_{\xi} = 84 $
    $ \xi = 0.001 $ $ \xi = 0.0001 $ $ \xi = 0.000001 $
    $ J_{\xi}^{*} = 2.867417 $ $ J_{\xi}^{*} = 2.861134 $ $ J_{\xi}^{*} = 2.860158 $
    $ M_{\xi} = 128 $ $ M_{\xi} = 206 $ $ M_{\xi} = 286 $
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